Ratio and Proportion
Ratio and Proportion Formulas
Definitions
Ratio: A ratio is a relationship between two numbers indicating how many times the first number contains the second.
Proportion: A proportion states that two ratios are equal.
Basic Formulas
Ratio
The ratio of \( a \) to \( b \) is written as:
\[ a : b \quad \text{or} \quad \frac{a}{b} \]Proportion
If \( a : b = c : d \), then \( a, b, c, \) and \( d \) are in proportion. It can be written as:
\[ \frac{a}{b} = \frac{c}{d} \]Properties of Ratios
Multiplication Property
If \( \frac{a}{b} = \frac{c}{d} \), then multiplying both sides by the same non-zero number gives the same ratio:
\[ \frac{a \times k}{b \times k} = \frac{c \times k}{d \times k} \]Inversion Property
If \( \frac{a}{b} = \frac{c}{d} \), then inverting both sides gives:
\[ \frac{b}{a} = \frac{d}{c} \]Equality of Ratios
If \( a : b = c : d \), then:
\[ a \times d = b \times c \]Properties of Proportions
Alternendo
\p>If \( \frac{a}{b} = \frac{c}{d} \), then: \[ \frac{a}{c} = \frac{b}{d} \]Componendo
If \( \frac{a}{b} = \frac{c}{d} \), then:
\[ \frac{a + b}{b} = \frac{c + d}{d} \]Dividendo
If \( \frac{a}{b} = \frac{c}{d} \), then:
\[ \frac{a - b}{b} = \frac{c - d}{d} \]Componendo and Dividendo
If \( \frac{a}{b} = \frac{c}{d} \), then:
\[ \frac{a + b}{a - b} = \frac{c + d}{c - d} \]Example Problems
Example 1: Finding Ratio
If the ratio of boys to girls in a class is 3:2, then for every 3 boys, there are 2 girls.
Example 2: Finding Proportion
If 4 pens cost $20 and 6 pens cost $30, then the cost of pens is in proportion.
\[ \frac{4}{20} = \frac{6}{30} \]Master This Topic with Smart Practice
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